The eldritch invocation Far Scribe lets a creature put its name in the book. Likewise for Gift of the Protectors.
With your permission, a creature can use its action to write its name on that page, which can contain a number of names equal to your proficiency bonus.
The nearest equivalent to this is the wizard's spell book, (indeed there's The Book of Ancient Secrets eldritch invocation gives a warlock the ability to cast ritual spells), but I don't think the Book of Shadows behaves exactly the same as a wizard's spellbook. For instance a wizard can have any number of backup spell books, but the warlock can't back up the contents of their Book of Shadows:
If you lose your Book of Shadows, you can perform a 1-hour ceremony to receive a replacement from your patron. This ceremony can be performed during a short or long rest, and it destroys the previous book. The book turns to ash when you die.
A wizard's spellbook also obviously hasn't got an equivalent to the names added by various eldritch invocations.
If I recreate the book, do I still have its previous contents, names from invocations and whatnot in there? Does it matter where the contents come from?
The accepted answer on When a Book of Shadows is replaced due to loss, are rituals that were inscribed in it lost? says that anything written by the warlock is lost by analogy with the wizard's spellook, but anything from the patron is preserved. The other answers have rulings (that have no official weight) from Jeremy Crawford and Mike Mearls that the rituals and cantrips are preserved. The answerer argues it is due to rare inks.
Given that the warlocks Book of Shadows is an eldritch item provided by the Patron, and not a wizards 'mundane' spellbook, do the names added by others (i.e. content not added using rare inks) have the same fate as the ritual spells? Do they get preserved or not?